ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso
136 7. Tripartite entanglement in three-mode Gaussian states modes 2 ′ and 3 ′ ′ T . The CM σs of the state of modes 1, 2 ′ , and 3 ′ is then written in the following block form: ′ T σs = ⎛ ⎝ σ1 ε12 ′ 0 εT 12 ′ σ2 ′ 0 0 0 σ3 ′ where mode 3 ′ is now disentangled from the others. Thus G 1|(23) τ (σ T s ) = G 1|2′ ′ T τ (σ ⎞ ⎠ , (7.42) s ) . (7.43) Moreover, the reduced CM σ12 ′ of modes 1 and 2′ defines a nonsymmetric GLEMS, Eq. (4.39), with Det σ1 = a 2 , 1 Det σ2 ′ = 3a 6 2 + 9 (a2 − 2) a2 + 25 − 1 , 1 Det σ12 ′ = 3a 2 2 − 9 (a2 − 2) a2 + 25 + 3 , and we have shown that the Gaussian contangle (and the whole family of Gaussian entanglement measures, Sec. 3.2.2) is computable in two-mode GLEMS, via Eq. (4.72). After some algebra, one finds the complete expression of Gres τ for T states: G res τ (σ T s ) = arcsinh 2 25R − 9a 4 + 3Ra 2 + 6a 2 − 109 − 81a 8 − 432a 6 + 954a 4 − 1704a 2 + 2125 − 3a 2 − 11 3a 2 − 7 3a 2 + 5 1 1 2 √ 2 R 2 × 18 3a 2 − R + 3 − 1 2 , (7.44) with R ≡ 9a 2 (a 2 − 2) + 25. What is remarkable about T states is that their tripartite Gaussian contangle, Eq. (7.44), is strictly smaller than the one of the GHZ/W states, Eq. (7.40), for any fixed value of the local mixedness a, that is, for any fixed value of the only parameter (operationally related to the squeezing of each single mode) that completely determines the CMs of both families of states up to local unitary operations. 18 This hierarchical behavior of the residual Gaussian contangle in the two classes of states is illustrated in Fig. 7.4. The crucial consequences of this result for the structure of the entanglement trade-off in Gaussian states will be discussed further in the next subsection. 18Notice that this result cannot be an artifact caused by restricting to pure Gaussian decompositions only in the definition Eq. (7.36) of the residual Gaussian contangle. In fact, for T states the relation Gres τ (σT s ) ≥ Eres τ (σT s ) holds due to the symmetry of the reduced two-mode states, and to the fact that the unitarily transformed state of modes 1 and 2 ′ is mixed and nonsymmetric.
7.3. Sharing structure of tripartite entanglement: promiscuous Gaussian states 137 G Τ res 4 3 2 1 0 1 2 3 4 5 a Figure 7.4. Plot, as a function of the single-mode mixedness a, of the tripartite residual Gaussian contangle Gres τ Eq. (7.40) in the CV GHZ/W states (dashed red line); in the T states Eq. (7.44) (solid blue line); and in 50 000 randomly generated mixed symmetric three-mode Gaussian states of the form Eq. (2.60) (dots). The GHZ/W states, that maximize any bipartite entanglement, also achieve maximal genuine tripartite quantum correlations, showing that CV entanglement distributes in a promiscuous way in symmetric Gaussian states. Notice also how all random mixed states have a nonnegative residual Gaussian contangle. This confirms the results presented in Ref. [GA10], and discussed in detail and extended in Sec. 7.2.1, on the strict validity of the CKW monogamy inequality for CV entanglement in three-mode Gaussian states. 7.3.3. Promiscuous continuous-variable entanglement sharing The above results, pictorially illustrated in Fig. 7.4, lead to the conclusion that in symmetric three-mode Gaussian states, when there is no bipartite entanglement in the two-mode reduced states (like in T states) the genuine tripartite entanglement is not enhanced, but frustrated. More than that, if there are maximal quantum correlations in a three-party relation, like in GHZ/W states, then the two-mode reduced states of any pair of modes are maximally entangled mixed states. These findings, unveiling a major difference between discrete-variable (mainly qubits) and continuous-variable systems, establish the promiscuous nature of CV entanglement sharing in symmetric Gaussian states [GA10]. Being associated with degrees of freedom with continuous spectra, states of CV systems need not saturate the CKW inequality to achieve maximum couplewise correlations, as it was instead the case for W states of qubits, Eq. (1.52). In fact, the following holds. ➢ Promiscuous entanglement in continuous-variable GHZ/W three-mode Gaussian states. Without violating the monogamy constraint Ineq. (6.2), pure symmetric three-mode Gaussian states are maximally three-way entangled and, at the same time, possess the maximum possible entanglement between any pair of modes in the corresponding two-mode reduced states. The two entanglements are mutually enhanced.
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136 7. Tripartite entanglement in three-mode Gaussian states<br />
modes 2 ′ and 3 ′ ′<br />
T . The CM σs of the state of modes 1, 2 ′ , and 3 ′ is then written in<br />
the following block form:<br />
′<br />
T<br />
σs =<br />
⎛<br />
⎝<br />
σ1 ε12 ′ 0<br />
εT 12 ′ σ2 ′ 0<br />
0 0 σ3 ′<br />
where mode 3 ′ is now disentangled from the others. Thus<br />
G 1|(23)<br />
τ (σ T s ) = G 1|2′<br />
′<br />
T<br />
τ (σ<br />
⎞<br />
⎠ , (7.42)<br />
s ) . (7.43)<br />
Moreover, the reduced CM σ12 ′ of modes 1 and 2′ defines a nonsymmetric GLEMS,<br />
Eq. (4.39), with<br />
Det σ1 = a 2 ,<br />
1<br />
<br />
Det σ2 ′ = 3a<br />
6<br />
2 + 9 (a2 − 2) a2 <br />
+ 25 − 1 ,<br />
1<br />
<br />
Det σ12 ′ = 3a<br />
2<br />
2 − 9 (a2 − 2) a2 <br />
+ 25 + 3 ,<br />
and we have shown that the Gaussian contangle (and the whole family of Gaussian<br />
entanglement measures, Sec. 3.2.2) is computable in two-mode GLEMS, via<br />
Eq. (4.72). After some algebra, one finds the complete expression of Gres τ for T<br />
states:<br />
G res<br />
τ (σ T s ) = arcsinh 2<br />
25R − 9a 4 + 3Ra 2 + 6a 2 − 109<br />
−<br />
<br />
81a 8 − 432a 6 + 954a 4 − 1704a 2 + 2125<br />
− 3a 2 − 11 3a 2 − 7 3a 2 + 5 1 1<br />
2 √ 2<br />
R 2<br />
× 18 3a 2 − R + 3 − 1<br />
2<br />
<br />
, (7.44)<br />
with R ≡ 9a 2 (a 2 − 2) + 25.<br />
What is remarkable about T states is that their tripartite Gaussian contangle,<br />
Eq. (7.44), is strictly smaller than the one of the GHZ/W states, Eq. (7.40), for any<br />
fixed value of the local mixedness a, that is, for any fixed value of the only parameter<br />
(operationally related to the squeezing of each single mode) that completely<br />
determines the CMs of both families of states up to local unitary operations. 18 This<br />
hierarchical behavior of the residual Gaussian contangle in the two classes of states<br />
is illustrated in Fig. 7.4. The crucial consequences of this result for the structure of<br />
the entanglement trade-off in Gaussian states will be discussed further in the next<br />
subsection.<br />
18Notice that this result cannot be an artifact caused by restricting to pure Gaussian decompositions<br />
only in the definition Eq. (7.36) of the residual Gaussian contangle. In fact, for T states<br />
the relation Gres τ (σT s ) ≥ Eres τ (σT s ) holds due to the symmetry of the reduced two-mode states,<br />
and to the fact that the unitarily transformed state of modes 1 and 2 ′ is mixed and nonsymmetric.
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